Difference between two types of meet in order-theory

Question

In order theory, some author defined meet of any two sets A and B as A $\wedge B= A \cap B$

and some other defined as A $\wedge B= \cup\{C \in X: C \subseteq A \cap B \}$. When we come across such situation?

Answer 1

If the lattice is a family of sets, including $A,B$ and $A \cap B$, then $A \wedge B = A \cap B$, but it is also $\bigcup\{C \in X: C\subseteq A\cap B\}$.
Why? Because $A \cap B \subseteq A\cap B$, whence $A \cap B$ is on of those sets $C$.
Hence $A\cap B \subseteq \bigcup\{C\in X:C\subseteq A\cap B\}$.
On the other hand, if it is clear that $\bigcup\{C\in X:C\subseteq A\cap B\} \subseteq A\cap B$. Therefore we have the equality.

Notice this is not always the case: it can happen that $A,B \in X$ but $A \cap B \notin X$.
In that case $A\cap B = \bigcup\{C\in X:C\subseteq A\cap B\}$.